The furore surrounding the departure of Michael Gove's adviser Dominic Cummings may have died down. But his parting shot – a 250-page tract on everything from genetics to school effectiveness and teacher quality – raised questions in many people's minds about who makes education policy, and how. Quizzed on Cummings's influence, the secretary of state responded robustly, quoting Margaret Thatcher: "Advisers advise; ministers decide." But how true is that?

Politicians may sign off on key changes, but in reality much key thinking on school and higher education policy is hammered out long before the minister decides. It takes place outside Westminster in thinktanks, which are funded by a range of private, corporate and public donors. These are technically independent research institutes that in reality lean towards one party or another. From here, ideas make their way into the party manifestos through an intricate network of advisers – frequently recruited from the thinktanks themselves.

According to Sam Freedman, who was at one of the most influential, the rightwing Policy Exchange, before becoming an adviser to Michael Gove in opposition and government, most of the ideas now being rolled out in schools started life in right-of-centre thinktanks before the 2010 election. Freedman, now director of research at Teach First, the organisation that recruits high-flying graduates into teaching, says: "The ideas behind structural reform, like free schools, had been around for some time and worked on in Policy Exchange; as were the ideas for reforming teacher training [through the introduction of School Direct]." He says recent changes to the curriculum and to assessment came out of Reform and Civitas, two more right-leaning thinktanks.

"Policy development is much easier to do in a thinktank," he says. "Parliamentarians and their staff don't have much capacity. There is too much day-to-day nonsense going on."

Equally influential with the current coalition is the liberal-leaning Centreforum, whose advisory body is chaired by the schools minister, David Laws. For the past year it has been researching changes to secondary school accountability, and lobbied successfully for a progress measure to replace the maths and English GCSE threshold as the key indicator for school performance.

Chris Paterson, senior researcher at Centreforum, says: "Everyone knew there was a problem with the current measures. We were keen to move away from the five A*-C measure and towards measuring progress between ages 11 and 16. We talked to focus groups and headteachers and did research into how the progress measures might work, which led to a seminar this summer. Following that we were pushing for the threshold measure to be dropped and a double weighting for English and maths in the final best eight subjects."

"I do think it may have been different if we hadn't been pushing on this particular issue," he says. "We are independent but have a good relationship with members of all parties."

Paterson says knowing the right people matter greatly when it comes to influencing policy. He says the role of thinktanks is to channel ideas to the people who make or influence policy. "Personal contacts are important."

So just how easy is it to penetrate the networks and how representative are they of the people who implement and experience the ideas? All three party education leads, Michael Gove, David Laws and Tristram Hunt, are white Oxbridge-educated men, as are many of the policy advisers.

Notable women on the Tory side before 2010 were Liz Truss, then deputy director at Reform and now a minister, and Rachel Wolf, who started the New Schools Network advising free schools, but now works in the US. Both were also Oxbridge educated. According to one DfE insider, the atmosphere in the education policy world today can be a bit "old boys' network".

Prof Becky Francis describes her move from King's College London to be director of education at the RSA – where she worked on developing Ofsted policy on satisfactory schools – as an "eye opener". "It is much more macho than the academic world. I was immediately struck by how much banter there was about sport, for example.

"I was automatically part of this world where there is an emphasis on the pace of new thinking," she says. "The pace and self-absorption may mean a lack of reflection about the sector itself. I also wonder if subconsciously women are not thought of as being as innovative as men."

Two-way traffic between the thinktanks, government and opposition may be partly to blame for the lack of diversity, says Rick Muir, of the left-of-centre IPPR, which has just completed a year-long commission into the future of higher education. "The thinktanks themselves are very male. We try to recruit from as wide a pool as we can, offer flexible employment to parents, and pay all interns a living wage, but we all have to recognise there is a long way to go. We have a responsibility because people from thinktanks are often recruited into private offices and provide the Spads [special advisers to the secretary of state] and policy advisers," Muir says.

Anastasia De Waal, head of family and education policy at Civitas, thinks the gender imbalance is simply down to the highly politicised nature of education policy: "There are plenty of women in education, even a female dominance in some areas, so policymaking could mirror the male leadership scenario in schools: where the relatively small proportion of men appear to take up a disproportionate number of headships. People making education policy are essentially in politics. The fact more are male simply reflects the dearth of women entering politics."

Which flags up a more troubling issue: the lack of a clear role for education professionals in policymaking. The Headteachers' Roundtable group of school leaders came together via Twitter to try to address this.

It has been developing its own qualifications framework while pushing for more decisions to be taken out of the political cycle and given to an independent expert body. No party will firmly commit to this idea, but according to Dale Bassett, who moved from Reform to take charge of public policy at the exam board AQA, politicians may be missing a trick. "In some ways the ideas coming from the professionals - like the Headteachers' Roundtable – are the most interesting at the moment.

"But the concept of including anyone who knows what they are talking about is not institutionalised. I see it more clearly from the outside. As a leading exam board we have significant expertise that we can use to inform policy, but that is sometimes quite ad-hoc and late in the process. Ideally we would be brought into politicians' strategic thinking early on."

Social media may be starting to break down the barriers, suggests Policy Exchange's head of education, Jonathan Simons. "Politicians and advisers are reading blogs, picking up knowledge and in some cases changing their minds on things," he says.

But progress is too slow for some. Ros McMullen, principal of the David Young Community Academy in Leeds and a founder member of the Headteachers' Roundtable, says: "Too often it seems ministers announce a policy without any clear working through and then adjustment and development come later, after the system has panicked and become destabilised. This isn't good."

The thinktanks are now gearing up for the 2015 general election and the general view is that there is less radical thinking than there was before 2010. But dividing lines are emerging, with Labour focusing on teacher quality, skills and the middle tier or local authority role and the Tories sticking to choice and diversity arguments.

"What I want to see is policymakers coming to those who have expertise and saying: 'This is the outcome we wish to achieve – how do we do it?'" says McMullen. "That would be far more effective and far less divisive than the current process. After all, we do all want the very best outcomes for young people."